Lotsa good questions and I think that you'll get some good advice here.
Equipment - if you're just starting out I think that it makes good sense to buy a complete "kit" from one of the major manufacturers. That way you get a good complete set of stuff, plus a good loading manual.
I have a lot of different brands of stuff and like it all but if you want me to go out on a limb I'd recommend the RCBS or Hornady kit for the beginner. Usually the starter kit comes with your choice of dies, of course you'd need .444 dies (and a shellholder).
Lapping.... do it with bullets loaded with lapping compound. You fire them at reduced velocities, so it's from the breech to the muzzle. DON'T worry about this yet - I highly recommend you start reloading with jacketed first, to get the hang of it. You can use any .429-.430" jacketed pistol bullets. Some of them would even be good game bullets in your rifle, generally those heavier than 240 grains (265 Hornady, 270 Speer, 300 Speer/Sierra/Nosler?, etc.).
When you get ready to do some cast bullets, check back in and report how it's going. Use .44 bullets, they are common between the .44 Mag and .444 rifle.
Used equipment - probably not a good idea for the beginner. The only used 'equipment' I would recommend would be different reloading manuals, and you should start looking for one or two that are different from the one that comes with your reloading kit. Can't have too much reference information. Can sometimes find used books for sale at gunshops and gun shows.
OK now the rest of you chime in....